On the sandpile group of 3×n twisted bracelets

The sandpile group of a graph is an abelian group that arises in several contexts, and a refinement of the number of spanning trees of the graph. It is a subtle isomorphism invariant of the graph and closely connected with the graph Laplacian matrix. In this paper, the abstract structures of the sandpile groups on 3×n twisted bracelets are determined and it is shown that the sandpile groups of those twisted bracelets are always isomorphic the direct sum of two or three cyclic groups.